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from z3 import *
import sys
variables = {}
def reachable(graph, node_reachable):
# check reachability to node_reachable
clauses = []
# at least one pebble can be gotted to node_reachble (from config or otherwise)
clauses.append(1 <= Sum(list(variables[node_reachable]["in"].values())))
return clauses
def build_model(graph, add_property, node_reachable, config=None, config_count=0):
clauses = []
for node in graph["v"]:
variables[node] = {"out":{}, "in":{}}
# creat symbolic variables for each edge in/out
for node in graph["v"]:
for edge in graph["e"][node]:
variables[node]["out"][edge] = Int(node+"_out_"+edge)
variables[edge]["in"][node] = Int(edge+"_in_"+node)
# add initial pebbles
variables[node]["in"]["config"] = Int(node+"_in_config")
if config is not None:
for node in graph["v"]:
clauses.append(variables[node]["in"]["config"] == config[node])
else:
sum_terms = []
for node in graph["v"]:
clauses.append(variables[node]["in"]["config"] >= 0)
sum_terms.append(variables[node]["in"]["config"])
clauses.append(Sum(sum_terms) == config_count)
# make edges linked
for node in graph["v"]:
for edge in graph["e"][node]:
clauses.append(variables[node]["out"][edge] == variables[edge]["in"][node])
# encode possible moves
# make sure we don't move too many total from pebble
for node in graph["v"]:
sum_out = Sum(list(variables[node]["out"].values()))
sum_in = Sum(list(variables[node]["in"].values()))
# The number of pebbles out from a node is no more than half the sum
# of all pebbles that are moved into a node
clauses.append( Implies(sum_in % 2 == 0, sum_out <= (sum_in/2 )) )
clauses.append( Implies(sum_in % 2 == 1, sum_out <= ((sum_in-1)/2 )) )
# make sure each move is a valid amount
for node in graph["v"]:
for in_var in variables[node]["in"].values():
clauses.append(in_var >= 0)
for out_var in variables[node]["out"].values():
sum_in = Sum(list(variables[node]["in"].values()))
clauses.append(
Implies(And(sum_in % 2 == 0, sum_in > 2),
And(0 <= out_var, out_var <= (sum_in/2 ))) )
clauses.append(
Implies(And(sum_in % 2 == 1, sum_in > 2),
And(0 <= out_var, out_var <= ((sum_in-1)/2 ))) )
# encode cycle lemma
# TODO get all cycles, and remove them?
for node in graph["v"]:
# if we along one edge, do not move back along it
for edge in graph["e"][node]:
in1 = variables[node]["in"][edge]
in2 = variables[edge]["in"][node]
clauses.append(Implies(in1 > 0, in2 == 0))
clauses += add_property(graph, node_reachable)
return And(clauses)
def is_sat(f, output=True):
s = Solver()
s.add(f)
if output:
print(f)
if s.check() == sat:
if output:
print("SAT")
model = s.model()
# print(model)
moves = []
for key in model:
if "_in_" in key.name() and 0<model[key].as_long():
parts = key.name().split("_in_")
if parts[1] != "config":
moves.append(parts[1] + " sends " + str(model[key])+" pebbles to " +parts[0])
moves.sort()
if output:
print("\n".join(moves))
return moves
if output:
print("UNSAT")
return None
def test_simple_path():
config = {
"a": 4,
"b": 0,
"c": 0,
}
graph = {
"v": ["a", "b", "c"],
"e": {
"a": ["b"],
"b": ["a", "c"],
"c": ["b"],
},
}
node_reachable = "c"
run_reachable_test_case(graph, config, node_reachable, True)
config = {
"a": 3,
"b": 0,
"c": 0,
}
run_reachable_test_case(graph, config, node_reachable, False)
config = {
"a": 5,
"b": 0,
"c": 0,
}
run_reachable_test_case(graph, config, node_reachable, True)
config = {
"a": 2,
"b": 1,
"c": 0,
}
run_reachable_test_case(graph, config, node_reachable, True)
def test_cycle():
config = {
"a": 4,
"b": 0,
"c": 0,
"d": 0,
}
graph = {
"v": ["a", "b", "c", "d"],
"e": {
"a": ["b", "d"],
"b": ["a", "c"],
"c": ["b", "d"],
"d": ["a", "c"],
},
}
node_reachable = "c"
run_reachable_test_case(graph, config, node_reachable, True)
def test_merge():
# a
# / \
# b c
# \ /
# d
# |
# e
graph = {
"v": ["a", "b", "c", "d", "e"],
"e": {
"a": ["b", "c"],
"b": ["a", "d"],
"c": ["a", "d"],
"d": ["b", "c", "e"],
"e": ["d"],
},
}
node_reachable = "e"
config = {
"a": 4,
"b": 1,
"c": 1,
"d": 0,
"e": 0,
}
run_reachable_test_case(graph, config, node_reachable, True)
config = {
"a": 4,
"b": 1,
"c": 0,
"d": 0,
"e": 0,
}
run_reachable_test_case(graph, config, node_reachable, False)
def test_lemke():
# /-x-\
# / | | \
# a-b c d
# \ \ | /
# e--f
# \/
# v
graph = {
"v": ["a", "b", "c", "d", "e", "f", "v", "x"],
"e": {
"a": ["b", "e", "x"],
"b": ["a", "f", "x"],
"c": ["x", "f"],
"d": ["x", "f"],
"e": ["a", "f", "v"],
"f": ["b", "c", "d", "e", "v"],
"v": ["e", "f"],
"x": ["a", "b", "c", "d"],
},
}
config = {
"a": 0,
"b": 0,
"c": 0,
"d": 0,
"e": 0,
"f": 0,
"v": 0,
"x": 8,
}
run_reachable_test_case(graph, config, "v", True)
config = {
"a": 1,
"b": 1,
"c": 1,
"d": 1,
"e": 0,
"f": 0,
"v": 0,
"x": 4,
}
run_reachable_test_case(graph, config, "v", True)
run_pebbling_number_test_case(graph, 7)
def run_reachable_test_case(graph, config, node_reachable, expected_is_sat):
variables = {}
z3_model = build_model(graph, reachable, node_reachable, config=config)
out = is_sat(z3_model, output=False)
if (expected_is_sat and not out) or (not expected_is_sat and out):
print(z3_model)
print(out)
print("failed!")
def run_pebbling_number_test_case(graph, count):
variables = {}
for node_reachable in graph["v"]:
# TODO model is wrong, we need to say there is not a configure in
# which this isn't possible. This is solvability?
z3_model = build_model(graph, reachable, node_reachable, config_count=count)
out = is_sat(z3_model, output=False)
if not out:
print("failed peb. number: ", count, node_reachable)
print(out)
def main():
test_simple_path()
test_cycle()
test_merge()
test_lemke()
if __name__ == '__main__':
main()
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